170 Prof. Ludwig Boltzmann on the 



may be written 



<S>(K,L, a, H) dKdLclGdK. 



The number of these doublets for which p lies betw r een p 

 and p + dp is 



<P.dKdLdGdR. d ?-+- { P0d ^=^dKdLdG.dR d ^ 



-i 



cr cr cr 



pi 



in which <r=-^. and I - is the time which elapses from 



dt ' cr 



j pi 



a peri-centre to an apo-centre; this latter is, therefore, a given 



r*po fin 

 function of K, L, G, H ; and SP" = <£> — is likewise a function 



/ J pi 

 of these four variables. Let us limit our attention to those 

 doublets for which (1) the last apsidal line of the central 

 orbit makes an angle lying between e and e + de with a straight 

 line drawn in the plane of the orbit parallel to a fixed plane; 

 (2) two planes, one normal to the central orbit and including 

 the direction of motion of the centre of gravity, and the other 

 parallel to a fixed straight line T, make angles between co and 

 a + dco with each other; and (3) the direction of motion of the 

 centre of gravity lies within a cone of given direction of axis 

 and of infinitely small angle d\. We have then to multiply 

 by deda)d\/167r s , and the number of doublets in unit volume 

 which fulfil these conditions is 



^.T^T-dKdLdGdB.dpdedcodX. . . (15) 



If we denote by g and g + dg, h and h + dh, k and k + dk, 

 the limits between which the velocity-components of the 

 centre of gravity of the doublets must lie (the coordinate axes 

 being rectangular and fixed), then 



G 2 dGdX=dgdhdk. 



Now keep g, h, and k constant, and place at tbe centre of the 

 shell a system of rectangular coordinates whose ^-axis is in 

 the direction of Gr ; denote the coordinates and velocity-com- 

 ponents of the nucleus relatively to this system by x x , y^ z ly 

 u ii v i> w d respectively, and transform these six variables in 

 K, L, H, p, e, a). We then introduce a second system of co- 

 ordinates, referred to which the coordinates and velocities of 

 the shell are x 2 , y^ z 2, u %> ^2? w 2, respectively. The s-axis of 

 this second system is to be taken normal to the plane of the 

 central motion, and its #-axis in the section of the plane of 



i, 



